Some new results on eigenvectors via dimension, diameter, and Ricci curvature

Advances in Mathematics

Volumn 155, Issue 1, 2000, Pages 98-153

  Bakry, Dominique ^a   Qian, Zhongmin ^b,c  

^a CNRS   (France)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

^c Shanghai Tiedao University   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034666932     PISSN: 00018708     EISSN: None     Source Type: Journal    
DOI: 10.1006/aima.2000.1932     Document Type: Article

References (52)

1. Bakry D. On Sobolev and logarithmic Sobolev inequalities for Markov semigroups. Elworthy K. D. New Trends in Stochastic Analysis. 1997;43-75 World Scientific, Singapore.
2. Bakry D., Emery M. Diffusions hypercontractives. Sém. de Prob. XIX. Lecture Notes in Math. 1123:1985;Springer-Verlag, Berlin. p. 177-206.
3. Bakry D., Ledoux M. Sobolev inequalities and Myers's diameter theorem for an abstract Markov generator. Duke Math. J. 85:1996;253-270.
4. Bakry D., Qian Z. On Harnack estimates for positive solutions of the heat equation on a complete manifold. C. R. Acad. Sci. Paris. 324:1997;1037-1042.
5. Bakry D., Qian Z. Harnack inequalities on a manifold with positive or negative Ricci curvature. Rev. Math. Iberoamericana. 15:1999;143-179.
6. Bérard P. H. Spectral Geometry: Direct and Inverse Problems. Lecture Notes in Mathematics. 1207:1986;Springer-Verlag, Berlin.
7. Berger M., Gauduchon P., Mazet M. Le Spectre d'une Variété Riemanniene. Lecture Notes in Mathematics. 194:1974;Springer-Verlag, Berlin.
8. Bochner S. Vector fields and Ricci curvature. Bull. Amer. Math. Soc. 52:1980;776-797.
9. 2/4. Geometry of the Laplace Operator. Proc. Symp. Pure Math. 36:1980;Am. Math. Soc. Providence. p. 29-77.
10. Buser P. A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. 15:1982;213-230.
11. Buser P. Geometry and Spectra of Compact Riemann Surfaces. 1992;Birkhäuser, Basel.
12. R. Brooks, Spectral Geometry, Cambridge Univ. Press, Cambridge, UK.
13. Cai K. R. Estimate on lower bound of the first eigenvalue of a compact Riemannian manifold. Chin. Ann. Math., Ser. B. 12:1991;267-271.
14. Chavel I. Eigenvalues in Riemannian Geometry. 1984;Academic Press, New York.
15. Chavel I. Riemannian Geometry: A Modern Introduction. Cambridge Tracts in Math. 108:1993;Cambridge Univ. Press, Cambridge.
16. Cheeger J. A Lower Bound for the Smallest Eigenvalue of the Laplacian, Problems in Analysis. Symposium in Honor of S. Bochner. 1970;Princeton Univ. Press, Princeton. p. 195-199.
17. Cheeger J., Gromov G., Taylor M. Finite propagation speed, kernel estimates for functions of Laplace operator, and the geometry of complete Riemannian manifolds. J. Diff. Geom. 17:1982;15-53.
18. Cheeger J., Ebin D. Comparison Theorems in Riemannian Geometry. 1975;North Holland, Amsterdam.
19. Chen M. F., Wang F. Y. General formula for lower bound of the first eigenvalue on Riemannian manifolds. Sci. China (Ser. A). 40:1997;384-394.
20. Chen R. Eigenvalue estimate on a compact Riemannian manifold. Amer. J. Math. 111:1989;769-781.
21. Chen R. Neumann eigenvalue estimate on a compact Riemannian manifold. Proc. Amer. Math. Soc. 108:1990;961-970.
22. Chen R., Li P. On Poincaré type inequalities. Trans. Amer. Math. Soc. 349:1997;1561-1585.
23. Cheng S. Y. Eigenvalue comparison theorems and its geometric applications. Math. Zeit. 143:1975;289-297.
24. Cheng S. Y., Li P. Heat kernel estimates and lower bounds of eigenvalues. Comment. Math. Helv. 56:1981;327-338.
25. Cheng S. Y., Yau S.-T. Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math. 28:1975;333-354.
26. e série. 13:1980;419-435.
27. Davies B. Heat Kernel and Spectral Theory. Cambridge Tracts in Math. 92. 1990;Cambridge Univ. Press, Cambridge.
28. Debiard A., Gaveau B., Mazet E. Théorèmes de comparaison en géométrie Riemannienne. Publ. RIMS, Kyoto Univ. 12:1976;391-425.
29. Chung F. R. K., Grigor'yan A., Yau S.-T. Upper bounds for eigenvalues of the discrete and continuous Laplace operators. Adv. Math. 117:1996;165-178.
30. Gallot S. Inégalités isopérimétriques et analytiques sur les variétés riemanniennes. Astérique. 163-164:1988;31-91.
31. Gilbarg D., Trudinger N. S. Elliptic Partial Differential Equations of Second Order. 1977;Springer-Verlag, Berlin.
32. M. Gromov, Paul-Levy's Isoperimetric Inequality, 1980.
33. Jia F. Estimate of the first eigenvalue on a compact Riemannian manifold with negative lower bound of Ricci curvature. Chin. Ann. Math. 12:1991;496.
34. Kröger P. On the spectral gap for compact manifolds. J. Diff. Geom. 36:1992;315-330.
35. Li P. A lower bound for the first eigenvalue of the Laplacian on a compact manifold. Indiana U. Math. J. 28:1979;1013-1019.
36. P. Li and S.-T. Yau, Eigenvalues of a compact Riemannian manifold, AMS Symposium on Geometry of the Laplace Operator, XXXVI, Hawaii, pp. 205-240, 1979.
37. Li P., Yau S.-T. On the parabolic kernel of the Schrödinger operator. Acta Math. 156:1986;153-201.
38. Lichnerowicz A. Géometries des Groupes des Transformations. 1958;Paris, Dunod.
39. Milnor J. Morse Theory. Ann. Math. Studies. 51:1963;Princeton Univ. Press, Princeton.
40. Myers S. B. Connections between differential geometry and topology. Duke Math. J. 1:1935;376-391.
41. Obata M. Certain conditions for a Riemannian manifold to be a sphere. J. Math. Soc. Japan. 14:1962;333-340.
42. Reid W. T. Ordinary Differential Equations. 1971;Wiley, New York.
43. Schoen R., Yau S.-T. Lectures on Differential Geometry. 1994;International Press.
44. Schoen R., Wolpert S., Yau S. T. Geometric bounds on the low eigenvalues of a compact surface. Geometry of the Laplace Operator. Proc. of Sym. in Pure Math. 36:1980;. p. 279-285.
45. Singer I. M., Wong B., Yau S. S. T., Yau S. T. An estimate of the gap of the first two eigenvalues of the Schrödinger operator. Ann. Scuola Norm. Pisa. 12:1985;319-333.
46. Swanson C. A. Comparison and Oscillation Theory of Linear Equations. 1968;Academic Press, New York.
47. Xu Y. The first nonzero eigenvalue of Neumann problem on Riemannian manifolds. J. Geometric Anal. 5:1995;151-165.
48. Yang H. C. Estimate of the first eigenvalue on a compact Riemannian manifold with negative lower bound of Ricci curvature. Sci. China (Ser. A). 32:1984;689.
49. Yau S.-T. Isoperimetric constants and the first eigenvalue of a compact manifold. Ann. Sci. Éc. Norm. Sup. 8:1975;487-507.
50. Yau S.-T. Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Math. J. 25:1976;659-670.
51. Yau S.-T. Nonlinear analysis in geometry. Enseign. Math. 33:1987;109-158.
52. Zhong J. Q., Yang H. C. On the estimate of the first eigenvalue of a compact Riemannian manifold. Sci. Sinica Ser. A. 27:1984;1265-1273.